Question

n,3n,5n........is an arithmetic sequence A). What is the common difference B). Prove that the sum of first n terms of this sequence is n3 C) find the sum of 15 terms is the sequence 15,45,75,........

Answer 1

Answer:

Step-by-step explanation:

The given sequence is n,3n,5n,......

that is, a₁ = n  ,  a₂ = 3n  ,  a₃ = 5n

A) Common difference = a₂-a₁

                                   = 3n-n

                                   = 2n

B) first term, a₁ = f = n

coomon difference,d = 2n

nth term = dn+(f-d)

             = (2n×n)+(n-2n)

             = 2n²+-n

             = 2n²-n

sum = $\frac{n[first term+nth term]}{2}$

        = n/2×[n+2n²-n]

        = n/2×[2n²+0]

        = n/2×[2n²]

        = n×n²

        = n³

C) Given sequence is 15,45,75,.....

f = 15

d = 45-15 = 30

nth term = dn+(f-d)

             = 30n+(15-30)

             = 30n-15

15th term = 30×15 -15

               = 450-15

               = 435

sum = n/2 [f+nth term]

       = 15/2 [15+435]

       = 15/2 [450]

       = 15×225

       = 3375